Related papers: Groups of order $p^4$ made less difficult

We examine which $p$-groups of order $\le p^6$ are Beauville. We completely classify them for groups of order $\le p^4$. We also show that the proportion of 2-generated groups of order $p^5$ which are Beauville tends to 1 as $p$ tends to…

Let $PU_n$ the projective unitary group of rank $n$ and $BPU_n$ its classifying space. For an odd prime $p$, we extend previous results to a compete description of $H^s(BPU_n;\mathbb{Z})_{(p)}$ for $s<2p+5$ by showing that the $p$-primary…

We consider groups of the nilpotency class $2$ of order $p^4$ which are the additive groups of local nearrings. It was shown that, for odd p, out of 6 of such groups 4 of them are the additive groups of local nearrings. Some examples of…

Generalising a previous result, we determine all non-abelian finite simple groups whose order has largest prime divisor not exceeding $10^4$. The computer code for this and similar calculations is made available.

A classification up to isomorphism of all left braces of order $p^3$, where $p$ is any prime number, is given. To this end, we first classify all the left braces of order $p$ and $p^2$, and then we construct explicitly the hypothesis…

A group of order $p^n$ ($p$ prime) has an indecomposable polynomial invariant of degree at least $p^{n-1}$ if and only if the group has a cyclic subgroup of index at most $p$ or it is isomorphic to one of two particular groups of small…

Let $N(n)$ denote the number of isomorphism types of groups of order $n$. We consider the integers $n$ that are products of at most $4$ not necessarily distinct primes and exhibit formulas for $N(n)$ for such $n$.

A group is called capable if it is a central factor group. For each prime $p$ and positive integer $c$, we prove the existence of a capable $p$-group of class $c$ minimally generated by an element of order $p$ and an element of order…

In this paper we present new examples of simple $p$-local compact groups for all odd primes. We also develop the necessary tools to show saturation, simpleness and the non-realizability as $p$-compact groups or compact Lie groups, which can…

The mininal degree of a finite group G, mu(G), is defined to be the smallest natural number n such that G embeds inside Sym(n). The group G is said to be exceptional if there exists a normal subgroup N such that mu(G/N)>mu(G). We will…

In this paper we classify all capable finite $p$-groups with derived subgroup of order $p$ and $G/G'$ of rank $n-1$.

For any abelian group $G$, we classify up to isomorphism all $G$-gradings on the classical central simple Lie algebras, except those of type $D_4$, over the field of real numbers (or any real closed field).

We study finite groups which possess a strongly p-embedded subgroup for some odd prime p. The main results of the paper will be applied in the ongoing project to classify the simple groups of local characteristic p.

It is proved that finite nonabelian simple groups $S$ with $\max \pi(S)=37$ are uniquely determined by their order and degree pattern in the class of all finite groups.

In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.

Recall that a $p$-group of order $p^ {n} >p^ {3} $ is of maximal class, if its nilpotency class is $n-1$. In this paper, we study the $p$-groups of maximal class. Furthermore, we introduce a subgroup of a $p$-group of maximal class called…

This paper presents a classification of all p-groups of order p^5 up to isomorphism. It contains a full list of their polycyclic presentations, a short introduction to the basic ideas of the methodes used to classify the groups, and a…

Let $P$ be a finite $p$-group and $p$ be an odd prime. Let $\mathcal{A}_p(P)_{\geq2}$ be a poset consisting of elementary abelian subgroups of rank at least 2. If the derived subgroup $P'\cong C_p\times C_p$, then the spheres occurring in…

This is the fourth one in a series of papers classifying the factorizations of almost simple groups with nonsolvable factors. In this paper we deal with orthogonal groups of minus type.

Given a prime number \(p\) and a natural number \(m\) not divided by \(p\), we propose the problem of finding the smallest number \(r_{0}\) such that for \(r\geq r_{0}\), every group \(G\) of order \(p^{r}m\) has a non-trivial normal…